After some digging:
1963 LA Dodgers; 9-24-63 clinching the pennant.
I narrowed it down to 1963 using the tape on the lockers, Roy Gleason only played 8 games with the Dodgers--all in 1963. No other Gleasons played in an applicable time period w/the other players. The strange part is that there's only one Miller on the team that year (Bob Miller). In the picture it's got him as #38 but baseball-almanac.com has listed as #15. There isn't any 38 listed for the '63 Dodgers as far as I can find.
My guess for the specific date--According to Retrosheet.org, Gleason played between 9-3-1963 and 9-28-1963; the Dodgers clinched on 9-24-06. Koufax's no-no that year was in May, so it can't be that.
From left to right:Sandy Koufax , Don Drysdale, Ron Perranoski, Pete Richert, Walter Alston, Lee Walls, Don Zimmer(???his # doesn't look like 23, but that was his # as Dodger???), and then the other three guys I can't get. I think the 2nd guy from the right (just a fraction of his face)

is Willie Davis, but I don't know.

The only ones I recognize are Drysdale, Koufax and the Duke.

The one to the right of Walt Alston looks like it is #37, Ed Roebuck. Here's a different picture of Roebuck for comparison:

http://www.angelfire.com/mb/markssigningbonus/edroebuck.JPG

3d guy from the right looks like junior gilliam to me -- he was on the team that year.

never mind -- now that i look at some other photos, that's not junior gilliam

If the photo is playoff time 1963, no Duke, no Zimmer and no Roebuck . . .

April 1, 1963: Sold Duke Snider to the New York Mets.

June 24, 1963: Sold Don Zimmer to the Washington Senators.

July 30, 1963: Traded Ed Roebuck to the Washington Senators. Received Marv Breeding.

I'm going to go with the pennant, but it's occuring prior to the game instead of after it.

The Cardinals lost that day and that gave the Dodgers the title, Alston was at the dentist when he heard, chances are he passed on the cake that day.

The one guy can't be Zimmer(maybe Pete Reiser?) Zim was already a Senator by September, perhaps the man on the far right is Doug Camilli?

Ed Roebuck and Snider were both no longer with the team, but Snider was visiting as a Met, the missing number is #27

Tommy Davis is in the pic behind the coach ...

I think that's Tommy Davis directly behind #37, and Willie Davis to his left.

The locker next to Gleason isn't Bob Miller, it's Joe Moeller (LAD '62, '64, '66-'71.

That might be Dick Tracewski on the far right. Trying to figure out who #37 is, there's a picture of the coaches here:

http://tinyurl.com/kk946

but I don't think he looks like any of them. Maybe it's the bullpen catcher.

Yeah, upon closer examination, the tag to the right locker does look like it says "Moeller", who was #38. Good catch Vern. So the problem is, Moeller didn't play for the Dodgers in '63. But Gleason didn't play for the Dodgers any other time than those 8 games I mentioned. Reading www.baseballlibrary.com article about Moeller, says he was plagued with injuries. Maybe he was called up but didn't play that year?

Sorry, but which guy is Duke Snider? I don't know the Dodgers at all, but it looks like Drysdale to me (to right of Koufax). Someone who knows better than me can correct me.

Moeller may have been a September call-up who just never got into a game.

Snider isn't in the picture.

The real challenge is going to be figuring our who the guy is peeking from behind the shoulder of the guy behind Koufax. (And the guy directly behind Koufax is going to be even tougher.) :)

Looking at Dodger coaches, they had four - Greg Mulleavy, Leo Durocher, Joe Becker and Pete Reiser. Reiser's height was 5'11, Alston was 6'2. I can't find anything on Joe Becker ... my bet, though, is Reiser.

As for the fellas on the very right - one is Willie Davis, the other could be either Dick Calmus or Ken McMullen ... ?

Mike, you're pretty sure about Willie Davis? Is he the 2nd from right guy?

Yeah, pretty sure ... wonder if the guy peweking behind Koufax is either Johnny Podres or Carroll Berringer, who was batting practice pitcher for them then?

My, My how technology has changed, on the Double Bubble box on the table they are offering *gasp* a free flashlight with batteries!!!

Zimmer would have been the shortest guy on the team at 5'9", and Don Drysdale the second tallest at 6'6" (and that's clearly him in back). But Zim can't be there if this is the clinch ... hmm. Have to do more thinking about this one.

Is it somebody's birthday during the 1963 Spring Training? Had Gleason yet begun his military service in S.T. '64? It looks like everyone pictured is singing HAPPY BIRTHDAY to whoever is standing next to the cameraman.

Also the CASH can is the kind of "semi-mandatory donation" thing that would appear on someone's birthday.

The Rev's comment is an interesting one. Clinches and suchlike are conventionally celebrated with champaigne, not cake. So I would posit that perhaps this times the Gleason locker tape artifact makes this a celebration of Alston completing ten years as the team's skipper. Alston is front and center; it's not his birthday (December 1), well into the offseason when we could be reasonably assured that the team wouldn't be together, and neither is it Koufax's (December 30) for the same reason. So the date is September 29, 1963, or thereabouts.

Another, perhaps less likely possibility: the place is the St. Louis locker room, the occaision is Walter Alston's 900th career victory (Gleason caught that game), and the date is September 16, 1963.

Regarding Redsauce's question about Joe Moeller, indeed, the Times says he was called up to the big club on September 20, 1963, but appeared in no games that year.

Other possibilities:

Doug Camilli's birthday (9/22)
Dick Nen's birthday (9/24)
Johnny Podres' birthday (9/30)

The Dodgers would have been home for games on all of those dates in 1963.

I can't find a record of Pete Reier's uniform number while he was a coach - but he was #27 in multiple seasons; he spent more seasons as 27 than as any other number. In particular, he was 27 in his first 3 full seasons with Brooklyn. Together with Mike's guess, above, based on height - I think it's increasingly likely that the man in uniform is Pete Reiser.

a follow-up: the Pete Reiser guess also agrees with the portion of his number visible in this 1963 team photo: http://walteromalley.com/phot_detail.php?gallery=2&set=36&photoID=141

The reason I sugegsted Spring Training is that the lockers seem a bit barren - Rob's suggestion of the St. Louis locker room might also suffice as a place with a less lived-in feel to it.

Good job everyone. The answers to Foto Friday #2 have been added beneath the photo above.

I had a great time with this, please keep this up! Great job by everyone and thanks for the help, I enjoyed this quite a bit.

I believe that's Ron Fairly on the far right next to Davis and the coach?

Someone suggested the coach might be pitching coach Joe Becker...No Way..Becker was much older and looked nothing like that guy.